1 / 19

Chapters 9

Chapters 9. Self-Similar Traffic. Introduction- Motivation. Validity of the queuing models we have studied depends on the Poisson nature of data traffic Recent studies show that Internet traffic patterns are often bursty and self-similar , not Poisson (exponential)

reeves
Download Presentation

Chapters 9

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapters 9 Self-Similar Traffic

  2. Introduction- Motivation • Validity of the queuing models we have studied depends on the Poisson nature of data traffic • Recent studies show that Internet traffic patterns are often bursty and self-similar, not Poisson (exponential) • Patterns appear through time • Understanding of the characteristics of self-similar traffic is essential to enable analysis of modern networks Chapter 9 – Self-Similar Traffic

  3. d dx f(x) = F(x) = e -x Introduction- Motivation Exponential Distribution Exponential Density E[X] = X = 1/ F(x) = Pr[Xx] = 1 – e-x Chapter 9 – Self-Similar Traffic

  4. Order Sierpinski triangle …can be found …in chaos. Chapter 9 – Self-Similar Traffic

  5. 72 144 72 328 144 72 72 ??? 0 72 216 288 648 720 864 936 ??? 432 216 216 0 216 648 864 Self-Similar Time Series Example 0 8 24 32 72 80 96 104 216 224 240 248 288 296 312 320 648 656 672 680 720 728 744 752 864 872 888 896 936 944 960 968 Chapter 9 – Self-Similar Traffic

  6. Relevance in Networking • Clustering (a.k.a. burstiness) is common in network traffic patterns • patterns are typically persistent through time • the clusters may themselves be clustered • Poisson traffic demonstrates clustering in short term, but smoothes over long term (known as the “memoryless” property) • During bursts due to clustering, queue sizes in switches/routers may build up more than the classical M/M/1 model predicts • impact on buffer sizes? Chapter 9 – Self-Similar Traffic

  7. Relevance in Networking Bottom line - The M/M/1 queuing model assumes that arrivals and service times are exponential and, therefore, “memoryless” … real network traffic patterns might not be memoryless, therefore the M/M/1 model might be “optimistic” Chapter 9 – Self-Similar Traffic

  8. E[x(at)] aH Var[x(at)] a2H Rx(at, as) a2H Continuous-Time Self-Similar Stochastic Processes • Continuous time: 0 t  • A stochastic process x(t) is statistically self-similar with parameter H (0.5  H  1), if for a  0: a -H x(at) has the same statistical properties as x(t) • In other words: • E[x(t)] = mean • Var[x(t)] = variance • Rx(t, s) = autocorrelation Chapter 9 – Self-Similar Traffic

  9. Hurst Parameter (H) • Measure of the persistence of a statistical phenomenon….. the measure of the long-range dependence of a stochastic process • H = 0.5 indicates absence of long-term dependence • As H approaches 1, the greater the degree of long-term dependence • Note: Brownian motion process B(t) is self-similar with H = 0.5 SEE Appendix 9A Chapter 9 – Self-Similar Traffic

  10. It is possible to define self-similar stochastic processes with heavy-tailed distributions • Leads to manageable distribution models • Can be used to characterize probability densities for traffic processes, e.g., packet interarrival times and burst lengths • Distribution of random variable X is heavy-tailed if: 1 – F(x) = Pr[X  x] ~ , as x  , 0   1 x  Heavy-Tailed Distributions Chapter 9 – Self-Similar Traffic

  11. k  x k +1 x  -1  k Pareto Heavy-Tailed Distribution • Characteristics: f(x) = F(x) = 0 (x  k) F(x) = 1 - (x  k,   0) f(x) = (x  k,   0) E[X] = k (  1) • Note that for k = 1, •  2, infinite variance •  1, infinite mean and variance Chapter 9 – Self-Similar Traffic

  12. +1  k kx f (x) = Pareto and Exponential Examples Density Functions Compared Chapter 9 – Self-Similar Traffic

  13. Examples of Self-Similar Data Traffic • LAN (Ethernet): • self-similar, H = 0.9 • Pareto fit for  = 1.2 • World-Wide Web: • browser traffic nicely fits Pareto distribution for 1.16    1.5 • file sizes on WWW seem to fit this distribution as well • TCP - FTP, TELNET: • session arrivals approximate Poisson • traffic patterns are bursty… heavy-tailed Chapter 9 – Self-Similar Traffic

  14. Mean Waiting Time (Delay) – Ethernet/ISDN Study Chapter 9 – Self-Similar Traffic

  15. Findings/Implications? • Higher loads lead to higher degrees of self-similarity • performance issues most relevant at high loads • Traditional Poisson modeling of traffic proven inadequate • leads to inaccurate queuing analysis results • increased delays, buffer size requirements • applicable to ATM, frame relay, 100BaseT switches, WAN routers, etc. • note excessive cell loss in first generation ATM switches • Pareto modeling yields better (more conservative) results Chapter 9 – Self-Similar Traffic

  16. Self-Similar Modeling (Norros) • Attempts to develop reliable analytical model of self-similar behavior • uses Fractional Brownian Motion (FBM) process (sect. 9.2) as basis • Buffer size can often be estimated using: q = 1/[2(1-H)] / (1-)H/(1-H) (note that for H = 0.5, this simplifies to /(1-), the M/M/1 model) Chapter 9 – Self-Similar Traffic

  17. Self-Similar Storage Model (Norros) Chapter 9 – Self-Similar Traffic

  18. Findings/Implications? • Buffer requirements much higher at lower levels of utilization for higher degrees of self-similarity (higher H) THEREFORE… • If higher levels of utilization are required, much larger buffers are needed for self-similar traffic than would be predicted using classical modeling So, what does all this really mean? Chapter 9 – Self-Similar Traffic

  19. Modeling and Estimating Self-Similarity • Task: determine if time series of data is actually is self-similar, and estimate the value of H • Variance Time Plot: Var[x(m)] vs. m • R/S Plot: log[R/S] vs. N • Periodogram: spectral density estimate • Whittle’s Estimator: estimate H, assuming self-similarity Chapter 9 – Self-Similar Traffic

More Related